Hybrid model

A hybrid model is a model that is used to model continuously discrete systems (also known as hybrid dynamic systems ).

There is no generally applicable definition for the term hybrid dynamic system. A hybrid dynamic system is characterized in that it interactively both continuous and discrete dynamics simultaneously to coordinate and control the location.

The basic feature of the hybrid models is the merging of continuous and discrete models. With the help of these hybrid models, switching dynamics can be described, as they often occur in process engineering, for example.

Example from process engineering

Dosing template system

The system consists of a number of storage containers for liquids (upstream before further processing stages). The containers empty over time at a constant rate (volume per time). If the liquid level in a container reaches a lower mark, it is filled by a server. Only one container can be filled at a time. The filling continues until the liquid level in one of the other containers has reached the lower mark. The filling process in the current container is canceled and the server switches the filling to the container to be refilled. (This example was first discussed by Chase, Serrano and Ramadge 1993 for three containers and by Schürmann and Hoffmann 1995 for any number of containers.)

An analysis shows that the dynamics of the system are structurally stable in sections. This means that no switchover occurs in certain time periods (in the example: change of server position). Within such a section the system shows an ordinary continuous dynamic , e.g. B. can be described by an ordinary differential equation . As soon as the server position changes, a switching of the system and the dynamics is observed. In general, complex systems then another differential equation describing the continuous dynamics. The modes of the system are linked to the various possible differential equations : The hybrid system adopts a defined mode when it is in a certain section with structurally stable, continuous dynamics. ${\ displaystyle m \ in M}$

Formally

A mathematical description of hybrid models that takes greater account of the continuous component is given below.

Designates the state vector for the equation of motion, the state vector of the algebraic equations , the vector of all external input variables and designates the possible modes of the system, then the dynamics of the system is described by the following hybrid model: with and . ${\ displaystyle {\ vec {x}}}$ ${\ displaystyle {\ vec {z}}}$ ${\ displaystyle {\ vec {u}}}$ ${\ displaystyle m \ in M}$ ${\ displaystyle {\ vec {f}} ({\ dot {\ vec {x}}}, {\ vec {x}}, {\ vec {z}}, {\ vec {u}}, m) = {\ vec {0}}}$ ${\ displaystyle {\ vec {x}} (t_ {0}) = {\ vec {x}} _ {0}}$ ${\ displaystyle {\ vec {z}} ({\ vec {x}} _ {0}, t_ {0}) = {\ vec {z}} _ {0}}$

The determination in which mode m the system is, can, for. B. be done by a condition / event system that models the discrete dynamics of the hybrid system, cf. e.g. Sreenivas and Krogh 1991.

literature

C. Chase, J. Serrano and PJ Ramadge: Periodicity and chaos from switched flow systems: contrasting examples of discretly controlled continuous systems . In: IEEE Trans. Automat. Contr. 38, 1993, pp. 70ff

T. Schürmann and I. Hoffmann: The entropy of strange billiards inside n-simplexes (PDF; 257 kB) . In: J. Phys. A28, 1995, page 5033ff, 1995, pp. 5033-5039

RS Sreenivas and BH Krogh: ON Condition / Event Systems with Discrete State Realization . In: Discrete Event Dynamic Systems: Theory and Application 1, 1991, pp. 209ff

Individual evidence

↑ http://num.math.uni-bayreuth.de/de/teaching/archive/ss_2006/01062/ausarbeit11.pdf | Uni Bayreuth, Hybrid MPC

[1] ttp://num.math.uni-bayreuth.de/de/teaching/archive/ss_2006/01062/ausarbeit11.pdf | Uni Bayreuth, Hybrid MPC